Abstract:
Let $d>0$ be square-free integer and $L_d$ be the Hilbert lattice,
i.e. the even lattice of signature (2, 2) such that
$L_d=\begin{pmatrix}0 & 1 \\1 & 0\\ \end{pmatrix} \oplus
\begin{pmatrix} 2 & 1\\1 & \frac{1-d}{2}\\ \end{pmatrix}$ when $d=1 \pmod{4}$,
or $L_d = \begin{pmatrix}0 & 1 \\1 & 0 \\ \end{pmatrix} \oplus
\begin{pmatrix} 2 & 0 \\0 & -2d \\ \end{pmatrix}$ when $d=2,3\pmod{4}$.
Consider $\Gamma_d=O^+(L_d)$ and denote by $A(\Gamma_d)$
the algebra of $\Gamma_d$-automorphic forms. The main goal of the
report is the following
Theorem: If the algebra $A(\Gamma_d)$ is free then $d \in \{2,3,5,6,13,21\}$.
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